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The compactness of a weakly singular integral operator on weighted Sobolev spaces. (English) Zbl 1323.46026

Summary: It is shown that the weakly singular integral operator \(\int_{-1}^{1}\big(\phi(\tau)/|\tau -t|^{\gamma}\big)\,d\tau\), where \(0< \gamma< 1\), maps the weighted Sobolev space \(W_{p;\alpha,\beta}^{(n)}(\Omega)\) compactly into itself for \(1< p< \infty\), \(0< \alpha+1/q, \beta+1/q< 1\) and \(n\in \mathbb{N}_0\).

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)

Software:

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Full Text: DOI Euclid

References:

[1] David Elliott and Susumu Okada, The finite Hilbert transform and weighted Sobolev spaces , Math. Nachr. 266 (2004), 34-47. · Zbl 1054.44004
[2] A. Kufner, Weighted Sobolev spaces , Wiley & Sons, New York, 1985. · Zbl 0567.46009
[3] S.G. Mikhlin and S. Prössdorf, Singular integral operators , Springer-Verlag, Berlin, 1986.
[4] F.W.J. Olver, et al., NIST Handbook of mathematical functions , NIST and Cambridge University Press, New York, 2010. · Zbl 1198.00002
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